Generalized benefit functions and measurement of utility

Luenberger [8] introduced the so-called benefit function that converts preferences into a numerical function that has some cardinal meaning. This measure has a number of remarkable properties and is a powerful tool in analyzing welfare issues ([10], [12], [13], [14]). This paper studies the conditions for a general function to make it a relevant welfare measure. Therefore, we introduce a large class of measures, called generalized benefit functions. The generalized benefit function is derived from the minimization of a convex function over the complement of a convex set. We show this class encompases as a special case the benefit function and is suitable to provide an alternative characterization of preferences. We also make a connection to the expenditure function through Fenchel duality theory and derive a duality result from Lemaire [7] for reverse convex optimization. Finally, we study the relationship between our class of functions and Hicksian compensated demand and we establish a link to the Slutsky matrix. Copyright Springer-Verlag 2004